Evolution of localized pulses in the defocusing modified Korteweg-de Vries equation theory

Abstract

In this work, we develop, in the Gurevich-Pitaevskii framework, an analytic theory for the evolution of localized pulses in the defocusing modified Korteweg-de Vries equation theory for situations when a dispersive shock does not eventually transform into a sequence of well-separated solitons. We found solutions to the Whitham modulation equations for the corresponding so-called "quasi-simple" dispersive shock waves and illustrated this solution with concrete examples of an initial pulse. Comparison of the analytical solution with direct numerical simulations showed that the modulation theory provides a very accurate description of the wave pattern even at one wavelength scale.

Figures (5)

• Figure 1: (a) Profile of a general quasi-simple pulse $u_0(x)$. (b) The inverse function of this initial condition defines two branches $\bar{x}_1(r)$ and $\bar{x}_2(r)$, where $r=u^2$.
• Figure 2: Sketch of the distribution of the square root of Riemann invariants for a DSW in the interval $[x_l, x_r]$. (a) When $a = u_m^2$ is constant throughout the evolution, then $x_l = x_m$ and the shock lies entirely within region $B$. Panels (b) and (c) represent a more general situation where $a < u_m^2$ at different stages: (b) before the soliton edge reaches the second branch $\bar{x}_2(r)$, the DSW is located in region $A$ to the left of $x_m$; (c) after it enters this branch, the shock requires two distinct analytical descriptions, one for region $A$ ($[x_l, x_m]$) and another for region $B$ ($[x_m, x_r]$).
• Figure 3: Evolution of a quasi-simple positive pulse for the mKdV equatio at times (a) $t=0$, (b) $t=5$, (c) $t=10$, (d) $t=15$, (e) $t=20$, and (f) $t=25$. The plots show a comparison between numerical results obtained via the split-step Fourier method (dashed red lines) and analytical solutions (solid blue lines) derived from Eq. \ref{['eq6']} using the Riemann invariants.
• Figure 4: Trajectories of the soliton edge $x_r(t)$ and the small-amplitude edge $x_l(t)$, determined by Eqs. \ref{['eq76']} and \ref{['eq77']}, respectively.
• Figure 5: Analytical and numerical solutions of the mKdV equation evolved from the initial distribution \ref{['eq57']} at $t = 20$. (a) Profiles of the Riemann invariants used to analytically construct the DSW distribution via Eq. \ref{['eq6']}. The resulting wave profiles are shown for (b) positive and (c) negative initial pulses, corresponding to the two signs in Eq. \ref{['eq57']}. Analytical results (solid blue lines) are compared with numerical solutions obtained via the split-step Fourier method (dashed red lines).